We explore quantitative descriptors that herald when a many-particle system in d-dimensional Euclidean space Rd approaches a hyperuniform state as a function of the relevant control parameter. We establish quantitative criteria to ascertain the extent of hyperuniform and nonhyperuniform distance-scaling regimes n terms of the ratio B/Aa, where A is "volume" coefficient and ABis"surface−area"coefficientassociatedwiththelocalnumbervariance\sigma^2(R)forasphericalwindowofradiusR.Tocomplementtheknowndirect−spacerepresentationofthecoefficientBintermsofthetotalcorrelationfunctionh({\bf r}),wederiveitscorrespondingFourierrepresentationintermsofthestructurefactorS({\bf k}),whichisespeciallyusefulwhenscatteringinformationisavailableexperimentallyortheoretically.Weshowthatthefree−volumetheoryofthepressureofequilibriumpackingsofidenticalhardspheresthatapproachastrictlyjammedstateeitheralongthestablecrystalormetastabledisorderedbranchdictatesthatsuchendstatesbeexactlyhyperuniform.UsingtheratioB/A,thehyperuniformityindexHandthedirect−correlationfunctionlengthscale\xi_c,westudythreedifferentexactlysolvablemodelsasafunctionoftherelevantcontrolparameter,eitherdensityortemperature,withendstatesthatareperfectlyhyperuniform.Weanalyzeequilibriumhardrodsand"sticky"hard−spheresystemsinarbitraryspacedimensiond$ as a function of density. We also examine low-temperature excited states of many-particle systems interacting with "stealthy" long-ranged pair interactions as the temperature tends to zero. The capacity to identify hyperuniform scaling regimes should be particularly useful in analyzing experimentally- or computationally-generated samples that are necessarily of finite size.